From c43048b752d6b797bc238ea16a42180449bf63f1 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Fri, 29 Aug 2014 15:46:49 +0000
Subject: [PATCH] A tentative chaptern on coinductive types.

Currently I hitted a major bug somewhere.
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+(*
+Coinductive Types and Predicates
+*)
+
+include "arithmetics/nat.ma".
+include "basics/types.ma".
+include "basics/lists/list.ma".
+
+(* The only primitive data types of Matita are dependent products and universes.
+So far every other user defined data type has been an inductive type. An
+inductive type is declared by giving its list of constructors (or introduction
+rules in the case of predicates). An inhabitant of an inductive type is obtained
+composing together a finite number of constructors and data of other types,
+according to the type of the constructors. Therefore all inhabitants of inductive
+types are essentially finite objects. Natural numbers, lists, trees, states of
+a DFA, letters of an alphabet are all finite and can be defined inductively.
+
+If you think of an inhabitant of an inductive type as a tree of a certain shape,
+whose nodes are constructors, the only trees can be represented are trees of
+finite height. Note, however, that it is possible to have trees of infinite
+width by putting in the argument of a constructor of a type I an enumeration
+of elements of I (e.g. ℕ → I). *)
+
+(* Example of an infinitely branching tree of elements of type A stored in
+the nodes: *)
+inductive infbrtree (A: Type[0]) : Type[0] ≝
+   Nil: infbrtree A
+ | Node: A → (ℕ → infbrtree A) → infbrtree A.
+
+(* Example: the tree of natural numbers whose root holds 0 and has as children
+   the leafs 1,2,3,… *)
+example infbrtree_ex ≝ Node ℕ 0 (λn. Node ℕ (S n) (λ_.Nil ℕ)).
+
+(*** Infinite data types via functions ***)
+
+(* In mathematics and less frequently in computer science there is the need to
+also represent and manipulate types of infinite objects. Typical examples are:
+sequences, real numbers (a special case of sequences), data streams (e.g. as
+read from a network interface), traces of diverging computations of a program,
+etc. One possible representation, used in mathematics since a long time, is
+to describe an infinite object by means of an infinite collection of
+approximations (also called observations). Often, the infinite collection can
+be structured in a sequence, identified as a function from the domain of natural
+numbers. *)
+
+(* Example 1: sequences of elements of type A *)
+definition seq : Type[0] → Type[0] ≝ λA:Type[0]. ℕ → A.
+
+(* Example 2: Real numbers as Cauchy sequences and their addition. *)
+(* First we axiomatize the relevant properties of rational numbers. *)
+axiom Q: Type[0].
+axiom Qdist: Q → Q → Q.
+axiom Qleq: Q → Q → Prop.
+interpretation "Qleq" 'leq x y = (Qleq x y).
+axiom transitive_Qleq: transitive … Qleq.
+axiom Qplus: Q → Q → Q.
+interpretation "Qplus" 'plus x y = (Qplus x y).
+axiom Qleq_Qplus:
+ ∀qa1,qb1,qa2,qb2. qa1 ≤ qb1 → qa2 ≤ qb2 → qa1 + qa2 ≤ qb1 + qb2.
+axiom Qdist_Qplus:
+ ∀qa1,qb1,qa2,qb2. Qdist (qa1 + qa2) (qb1 + qb2) ≤ Qdist qa1 qb1 + Qdist qa2 qb2.
+axiom Qhalve: Q → Q.
+axiom Qplus_Qhalve_Qhalve: ∀q. Qhalve q + Qhalve q = q.
+
+(* A sequence of rationals. *)
+definition Qseq: Type[0] ≝ seq Q.
+
+(* The Cauchy property *)
+definition Cauchy: Qseq → Prop ≝
+ λR:Qseq. ∀eps. ∃n. ∀i,j. n ≤ i → n ≤ j → Qdist (R i) (R j) ≤ eps.
+
+(* A real number is an equivalence class of Cauchy sequences. Here we just
+   define the carrier, omitting the necessary equivalence relation for the
+   quotient. *) 
+record R: Type[0] ≝
+ { r: Qseq
+ ; isCauchy: Cauchy r
+ }. 
+
+(* The following coercion is used to write r n to extract the n-th approximation
+ from the real number r *)
+coercion R_to_fun : ∀r:R. ℕ → Q ≝ r on _r:R to ?. 
+
+(* Adding two real numbers just requires pointwise addition of the 
+ approximations. The proof that the resulting sequence is Cauchy is the standard
+ one: to obtain an approximation up to eps it is necessary to approximate both
+ summands up to eps/2. The proof that the function is well defined w.r.t. the
+ omitted equivalence relation is also omitted. *)
+definition Rplus: R → R → R ≝
+ λr1,r2. mk_R (λn. r1 n + r2 n) ….
+ #eps
+ cases (isCauchy r1 (Qhalve eps)) #n1 #H1
+ cases (isCauchy r2 (Qhalve eps)) #n2 #H2
+ %{(max n1 n2)} #i #j #K1 #K2 @(transitive_Qleq … (Qdist_Qplus …))
+ <(Qplus_Qhalve_Qhalve eps) @Qleq_Qplus [@H1 @le_maxl | @H2 @le_maxr]
+ [2,6: @K1 |4,8: @K2]
+qed.
+
+(* Example 3: traces of a program *)
+(* Let us introduce a very simple programming language whose instructions
+   can test and set a single implicit variable. *)
+inductive instr: Type[0] ≝
+   p_set: ℕ → instr                 (* sets the variable to a constant *)
+ | p_incr: instr                    (* increments the variable *)
+ | p_while: list instr → instr.     (* repeats until the variable is 0 *)
+
+(* The status of a program as the values of the variable and the list of
+ instructions to be executed. *)
+definition state ≝ ℕ × (list instr).
+
+(* The transition function from a state to the next one. *)
+inductive next: state → state → Prop ≝
+   n_set: ∀n,k,o. next 〈o,(p_set n)::k〉 〈n,k〉
+ | n_incr: ∀k,o. next 〈o,p_incr::k〉 〈S o,k〉
+ | n_while_exit: ∀b,k. next 〈0,(p_while b)::k〉 〈0,k〉
+ | n_while_loop: ∀b,k,o. next 〈S o,(p_while b)::k〉 〈S o,b@(p_while b)::k〉.
+
+(* A diverging trace is a sequence of states such that the n+1-th state is
+ obtained executing one step from the n-th state *)
+record div_trace: Type[0] ≝
+ { div_tr: seq state
+ ; div_well_formed: ∀n. next (div_tr n) (div_tr (S n))
+ }.
+
+(* The previous definition of trace is not very computable: we cannot write
+ a program that given an initial state returns its trace. To write that function,
+ we first write a computable version of next, called step. *)
+definition step: state → option state ≝
+ λs. let 〈o,k〉 ≝ s in
+  match k with
+   [ nil ⇒ None ?
+   | cons hd k ⇒
+      Some … match hd with
+      [ p_set n ⇒ 〈n,k〉
+      | p_incr ⇒ 〈S o,k〉
+      | p_while b ⇒
+         match o with
+         [ O ⇒ 〈0,k〉
+         | S p ⇒ 〈S p,b@(p_while b)::k〉 ]]].
+
+theorem step_next: ∀o,n. step o = Some … n → next o n.
+ * #o * [ #n normalize #abs destruct ]
+ * normalize
+ [ #n #tl * #n' #tl'
+ | #tl * #n' #tl'
+ | #b #tl * #n' #tl' cases o normalize [2: #r]]
+ #EQ destruct //
+qed.
+
+theorem next_step: ∀o,n. next o n → step o = Some … n.
+ * #o #k * #n #k' #H inversion H normalize
+ [ #v #tl #n'
+ | #tl #n'
+ | #b #tl]
+ #EQ1 #EQ2 //
+qed.
+  
+(* Termination is the archetipal undecidable problem. Therefore there is no
+ function that given an initial state returns the diverging trace if the program
+ diverges or fails in case of error. The best we can do is to give an alternative
+ definition of trace that captures both diverging and converging computations. *)
+record trace: Type[0] ≝
+ { tr: seq (option state)
+ ; well_formed: ∀n,s. tr n = Some … s → tr (S n) = step s
+ }.
+
+(* The trace is diverging if every state is not final. *)
+definition diverging: trace → Prop ≝
+ λt. ∀n. tr t n ≠ None ?.
+
+(* The two definitions of diverging traces are equivalent. *)
+theorem div_trace_to_diverging_trace:
+ ∀d: div_trace. ∃t: trace. diverging t ∧ tr t 0 = Some … (div_tr d 0).
+ #d %{(mk_trace (λn.Some ? (div_tr d n)) …)}
+ [2: % // #n % #abs destruct
+ | #n #s #EQ destruct lapply (div_well_formed d n) /2 by div_well_formed, next_step/ ]
+qed.
+
+theorem diverging_trace_to_div_trace:
+ ∀t: trace. diverging t → ∃d: div_trace. tr t 0 = Some … (div_tr d 0).
+ #t #H %
+ [ % [ #n lapply (H n) -H cases (tr t n) [ * #abs cases (abs …) // ] #s #_ @s
+ | #n  lapply (well_formed t n)
+   lapply (H n) cases (tr t n) normalize [ * #abs cases (abs …) // ]
+   * #o #k #_ lapply (H (S n)) -H
+   cases (tr t (S n)) normalize
+   [ * #abs cases (abs …) // ] * #o' #k' #_ #EQ lapply (EQ … (refl …)) -EQ
+     normalize cases k normalize [ #abs destruct ] #hd #tl #EQ destruct -EQ
+     @step_next >e0 // ]
+ | lapply (H O) -H cases (tr t O) [ * #abs cases (abs …) // ] // ]
+qed.
+
+(* However, given an initial state we can always compute a trace.
+   Note that every time the n-th element of the trace is accessed, all the
+   elements in position ≤ n are computed too. *)
+let rec compute_trace_seq (s:state) (n:nat) on n : option state ≝
+ match n with
+ [ O ⇒ Some … s
+ | S m ⇒
+    match compute_trace_seq s m with
+    [ None ⇒ None …
+    | Some o ⇒ step o ]].
+
+definition compute_trace: ∀s:state. Σt:trace. tr t 0 = Some … s.
+ #s %
+ [ %{(compute_trace_seq s)}
+   #n #o elim n [ whd in ⊢ (??%? → ??%?); #EQ destruct // ]
+   -n #n #_ #H whd in ; whd in ⊢ (??%?); >H //
+ | // ]
+qed.
+
+(*** Infinite data types as coinductive types ***)
+
+(* All the previous examples were handled very easily via sequences
+ declared as functions. A few critics can be made to this approach though:
+ 1. the sequence allows for random access. In many situations, however, the
+    elements of the sequence are meant to be read one after the other, in
+    increasing order of their position. Moreover, the elements are meant to be
+    consumed (read) linearly, i.e. just once. This suggests a more efficient
+    implementation where sequences are implemented with state machines that
+    emit the next value and enter a new state every time they are read. Indeed,
+    some programming languages like OCaml differentiate (possibly infinite)
+    lists, that are immutable, from mutable streams whose only access operation
+    yields the head and turns the stream into its tail. Data streams read from
+    the network are a typical example of streams: the previously read values are
+    not automatically memoized and are lost if not saved when read. Files on
+    disk are also usually read via streams to avoid keeping all of them in main
+    memory. Another typical case where streams are used is that of diverging
+    computations: in place of generating at once an infinite sequence of values,
+    a function is turned into a stream and asking the next element of the stream
+    runs one more iteration of the function to produce the next output (often
+    an approximation).
+ 2. if a sequence computes the n-th entry by recursively calling itself on every
+    entry less than n, accessing the n-th entry requires re-computation of all
+    entries in position ≤ n, which is very inefficient. 
+ 3. by representing an infinite object as a collection of approximations, the
+    structure of the object is lost. This was not evident in the previous
+    examples because in all cases the intrinsic structure of the datatype was
+    just that of being ordered and sequences capture the order well. Imagine,
+    however, that we want to represent an infinite binary tree of elements of
+    type A with the previous technique. We need to associate to each position
+    in the infinite tree an element of type A. A position in the tree is itself
+    a path from the root to the node of interest. Therefore the infinite tree
+    is represented as the function (ℕ → 𝔹) → A where 𝔹 are the booleans and the
+    tree structure is already less clear. Suppose now that the binary tree may
+    not be full, i.e. some nodes can have less than two children. Representing
+    such a tree is definitely harder. We may for example use the data type
+    (N → 𝔹) → option A where None would be associated to the position
+    b1 ... bn if a node in such position does not exist. However, we would need
+    to add the invariant that if b1 ... bn is undefined (i.e. assigned to None),
+    so are all suffixes b1 ... bn b_{n+1} ... b_{n+j}.
+
+ The previous observations suggest the need for primitive infinite datatypes
+ in the language, usually called coinductive types or codata. Matita allows
+ to declare coinductive type with the same syntax used for inductive types,
+ just replacing the keywork inductive with coinductive. Semantically, the two
+ declarations differ because a coinductive type also contains infinite
+ inhabitants that respect the typechecking rules.
+*)
+
+(* Example 1 revisited: non terminated streams of elements of type A *)
+coinductive streamseq (A: Type[0]) : Type[0] ≝
+ sscons: A → streamseq A → streamseq A.
+
+(* Coinductive types can be inhabited by infinite data using coinductive
+   definitions, introduced by the keyword let corec. The syntax of let corec
+   definitions is the same of let rec definitions, but for the declarations
+   of the recursive argument. While let rec definitions are recursive definition
+   that are strictly decreasing on the recursive argument, let corec definitions
+   are productive recursive definitions. A recursive definition is productive
+   if, when fully applied to its arguments, it reduces in a finite amount of time
+   to a constructor of a coinductive type.
+   
+   Let's see some simple examples of coinductive definitions of corecursive
+   streamseqs. *)
+
+(* The streamseq 0 0 0 ...
+   Note that all_zeros is not a function and does not have any argument.
+   The definition is clearly productive because it immediately reduces to
+   the constructor sscons. *)
+let corec all_zeros: streamseq nat ≝ sscons nat 0 all_zeros.
+
+(* The streamseq n (n+1) (n+2) ...
+   The definition behaves like an automaton with state n. When the
+   streamseq is pattern matched, the current value n is returned as head
+   of the streamseq and the tail of the streamseq is the automaton with
+   state (S n). Therefore obtaining the n-th tail of the stream requires O(n)
+   operation, but every further access to its value only costs O(1). Moreover,
+   in the future the implementation of Matita could automatically memoize
+   streams so that obtaining the n-th element would also be an O(1) operation
+   if the same element was previously accessed at least once. This is what
+   is currently done in the implementation of the Coq system for example.
+*)
+let corec from_n (n:ℕ) : streamseq nat ≝ sscons … n (from_n (S n)).
+
+(* In order to retrieve the n-th element from a streamseq we can write a
+   function recursive over n. *)
+let rec streamseq_nth (A: Type[0]) (s: streamseq A) (n:ℕ) on n : A ≝
+ match s with [ sscons hd tl ⇒
+  match n with [ O ⇒ hd | S m ⇒ streamseq_nth … tl m ]]. 
+
+(* Any sequence can be turned into the equivalent streamseq and the other
+   way around. *)
+let corec streamseq_of_seq (A: Type[0]) (s: seq A) (n:ℕ) : streamseq A ≝
+ sscons … (s n) (streamseq_of_seq A s (S n)).
+
+lemma streamseq_of_seq_ok:
+ ∀A:Type[0]. ∀s: seq A. ∀m,n.
+  streamseq_nth A (streamseq_of_seq … s n) m = s (m+n).
+ #A #s #m elim m normalize //
+qed.
+
+definition seq_of_streamseq: ∀A:Type[0]. streamseq A → seq A ≝ streamseq_nth.
+
+lemma seq_of_streamseq_ok:
+ ∀A:Type[0]. ∀s: streamseq A. ∀n. seq_of_streamseq … s n = streamseq_nth … s n.
+ //
+qed.
+
+(* Example 2 revisited: Real numbers as Cauchy sequences and their addition.
+   We closely follow example 2 replacing sequences with streamseqs.
+*)
+
+definition Qstreamseq: Type[0] ≝ streamseq Q.
+
+definition Qstreamseq_nth ≝ streamseq_nth Q.
+
+(* The Cauchy property *)
+definition Cauchy': Qstreamseq → Prop ≝
+ λR:Qstreamseq. ∀eps. ∃n. ∀i,j. n ≤ i → n ≤ j → Qdist (Qstreamseq_nth R i) (Qstreamseq_nth R j) ≤ eps.
+
+(* A real number is an equivalence class of Cauchy sequences. Here we just
+   define the carrier, omitting the necessary equivalence relation for the
+   quotient. *) 
+record R': Type[0] ≝
+ { r': Qstreamseq
+ ; isCauchy': Cauchy' r'
+ }. 
+
+(* The following coercion is used to write r n to extract the n-th approximation
+ from the real number r *)
+coercion R_to_fun' : ∀r:R'. ℕ → Q ≝ (λr. Qstreamseq_nth (r' r)) on _r:R' to ?. 
+
+(* Pointwise addition over Qstreamseq defined by corecursion.
+   The definition is productive because, when Rplus_streamseq is applied to
+   two closed values of type Qstreamseq, it will reduce to sscons. *)
+let corec Rplus_streamseq (x:Qstreamseq) (y:Qstreamseq) : Qstreamseq ≝
+ match x with [ sscons hdx tlx ⇒
+ match y with [ sscons hdy tly ⇒
+  sscons … (hdx + hdy) (Rplus_streamseq tlx tly) ]].
+
+(* The following lemma was for free using sequences. In the case of streamseqs
+ it must be proved by induction over the index because Qstreamseq_nth is defined by
+ recursion over the index. *)
+lemma Qstreamseq_nth_Rplus_streamseq:
+ ∀i,x,y.
+  Qstreamseq_nth (Rplus_streamseq x y) i = Qstreamseq_nth x i + Qstreamseq_nth y i.
+ #i elim i [2: #j #IH] * #xhd #xtl * #yhd #ytl // normalize @IH
+qed.
+
+(* The proof that the resulting sequence is Cauchy is exactly the same we
+ used for sequences, up to two applications of the previous lemma. *)
+definition Rplus': R' → R' → R' ≝
+ λr1,r2. mk_R' (Rplus_streamseq (r' r1) (r' r2)) ….
+ #eps
+ cases (isCauchy' r1 (Qhalve eps)) #n1 #H1
+ cases (isCauchy' r2 (Qhalve eps)) #n2 #H2
+ %{(max n1 n2)} #i #j #K1 #K2
+ >Qstreamseq_nth_Rplus_streamseq >Qstreamseq_nth_Rplus_streamseq
+ @(transitive_Qleq … (Qdist_Qplus …))
+ <(Qplus_Qhalve_Qhalve eps) @Qleq_Qplus [@H1 @le_maxl | @H2 @le_maxr]
+ [2,6: @K1 |4,8: @K2]
+qed.
+
+(***** Intermezzo: the dynamics of coinductive data ********)
+
+(* Let corec definitions, like let rec definitions, are a form of recursive
+ definition where the definiens occurs in the definiendum. Matita compares
+ types up to reduction and reduction always allows the expansion of non recursive
+ definitions. If it also allowed the expansion of recursive definitions, reduction
+ could diverge and type checking would become undecidable. For example,
+ we defined all_zeros as "sscons … 0 all_zeros". If the system expanded all
+ occurrences of all_zeros, it would expand it forever to
+ "sscons … 0 (sscons … 0 (sscons … 0 …))".
+ 
+ In order to avoid divergence, recursive definitions are only expanded when a
+ certain condition is met. In the case of a let-rec defined function f that is
+ recursive on its n-th argument, f is only expanded when it occurs in an
+ application (f t1 ... tn ...) and tn is (the application of) a constructor.
+ Termination is guaranteed by the combination of this restriction and f being
+ guarded by destructors: the application (f t1 ... tn ...) can reduce to a term
+ that contains another application (f t1' ... tn' ...) but the size of tn'
+ (roughly the number of nested constructors) will be smaller than the size of tn
+ eventually leading to termination.
+
+ Dual restrictions are put on let corec definitions. If f is a let-rec defined
+ term, f is only expanded when it occurs in the term "match f t1 ... tn with ...".
+ To better see the duality, that is not syntactically perfect, note that: in
+ the recursive case f destructs terms and is expanded only when applied to a
+ constructor; in the co-recursive case f constructs terms and is expanded only
+ when it becomes argument of the match destructor. Termination is guaranteed
+ by the combination of this restriction and f being productive: the term
+ "match f t1 ... tn ... with" will reduce in a finite amount of time to
+ a match applied to a constructor, whose reduction can expose another application
+ of f, but not another "match f t1' .. tn' ... width". Therefore, since no
+ new matches around f can be created by reduction, the number of
+ destructors that surrounds the application of f decreases at every step,
+ eventually leading to termination.
+ 
+ Even if a coinductively defined f does not reduce in every context to its
+ definiendum, it is possible to prove that the definiens is equal to its
+ definiendum. The trick is to prove first an eta-expansion lemma for the
+ inductive type that states that an inhabitant of the inductive type is
+ equal to the one obtained destructing and rebuilding it via a match. The proof
+ is simply by cases over the inhabitant. Let's see an example. *)
+
+lemma eta_streamseq:
+ ∀A:Type[0]. ∀s: streamseq A.
+  match s with [ sscons hd tl ⇒ sscons … hd tl ] = s.
+ #A * //
+qed.
+
+(* In order to prove now that the definiens of all_zeros is equal to its
+ definiendum, it suffices to rewrite it using the eta_streamseq lemma in order
+ to insert around the definiens the match destructor that triggers its
+ expansion. *)
+lemma all_zeros_expansion: all_zeros = sscons … 0 all_zeros.
+ <(eta_streamseq ? all_zeros) in ⊢ (??%?); //
+qed.
+
+(* Expansions lemmas as the one just presented are almost always required to
+ progress in non trivial proofs, as we will see in the next example. Instead
+ the equivalent expansions lemmas for let-rec definitions are rarely required.
+*)
+
+(* Example 3 revisited: traces of a program. *)
+
+(* A diverging trace is a streamseq of states such that the n+1-th state is
+ obtained executing one step from the n-th state. The definition of
+ div_well_formed' is the same we already used for sequences, but on
+ streamseqs. *)
+
+definition div_well_formed' : streamseq state → Prop ≝
+ λs: streamseq state.
+  ∀n. next (streamseq_nth … s n) (streamseq_nth … s (S n)). 
+
+record div_trace': Type[0] ≝
+ { div_tr':> streamseq state
+ ; div_well_formed'': div_well_formed' div_tr' 
+ }.
+
+(* The well formedness predicate over streamseq can also be redefined using as a
+ coinductive predicate. A streamseq of states is well formed if the second
+ element is the next of the first and the stream without the first element
+ is recursively well formed. *)
+coinductive div_well_formed_co: streamseq state → Prop ≝
+ is_next:
+  ∀hd1:state. ∀hd2:state. ∀tl:streamseq state.
+   next hd1 hd2 → div_well_formed_co (sscons … hd2 tl) →
+    div_well_formed_co (sscons … hd1 (sscons … hd2 tl)).
+
+(* Note that Matita automatically proves the inversion principles for every
+ coinductive type, but no general coinduction principle. That means that
+ the elim tactic does not work when applied to a coinductive type. Inversion
+ and cases are the only ways to eliminate a coinductive hypothesis. *)
+
+(* A proof of div_well_formed cannot be built stacking a finite
+ number of constructors. The type can only be inhabited by a coinductive
+ definition. As an example, we show the equivalence between the two
+ definitions of well formedness for streamseqs. *)
+ 
+(* A div_well_formed' stream is also div_well_formed_co. We write the proof
+ term explicitly, omitting the subterms that prove "next hd1 hd2" and
+ "div_well_formed' (sscond … hd2 tl)". Therefore we will obtain two proof
+ obligations. The given proof term is productive: if we apply it to a closed
+ term of type streamseq state and a proof that it is well formed, the two
+ matches in head position will reduce and the lambda-abstraction will be
+ consumed, exposing the is_next constructor. *)
+ 
+let corec div_well_formed_to_div_well_formed_co
+ (s: streamseq state): div_well_formed' s → div_well_formed_co s ≝
+ match s with [ sscons hd1 tl1 ⇒
+  match tl1 with [ sscons hd2 tl ⇒
+   λH: div_well_formed' (sscons … hd1 (sscons … hd2 tl)).
+    is_next … (div_well_formed_to_div_well_formed_co (sscons … hd2 tl) …) ]].
+[ (* First proof obligation: next hd1 hd2 *) @(H 0)
+| (* Second proof obligation: div_well_formed' (sscons … hd2 tl) *) @(λn. H (S n)) ]
+qed.  
+
+(* A div_well_formed_co stream is also div_well_formed'. This time the proof is
+ by induction over the index and inversion over the div_well_formed_co
+ hypothesis. *)
+theorem div_well_formed_co_to_div_well_formed:
+ ∀s: streamseq state. div_well_formed_co s → div_well_formed' s.
+ #s #H #n lapply H -H lapply s -s elim n [2: #m #IH]
+ * #hd1 * #hd2 #tl normalize #H inversion H #hd1' #hd2' #tl' #Hnext #Hwf
+ #eq destruct /2/
+qed.
+
+(* Like for sequences, because of undecidability of termination there is no
+ function that given an initial state returns the diverging trace if the program
+ diverges or fails in case of error. We need a new data type to represent a
+ possibly infinite, possibly terminated stream of elements. Such data type is
+ usually called stream and can be defined elegantly as a coinductive type. *)
+coinductive stream (A: Type[0]) : Type[0] ≝
+   snil: stream A
+ | scons: A → stream A → stream A.
+
+(* The definition of trace based on streams is more natural than that based
+ on sequences of optional states because there is no need of the invariant that
+ a None state is followed only by None states (to represent a terminated
+ sequence). Indeed, this is the first example where working with coinductive
+ types seems to yield advantages in terms of simplicity of the formalization.
+ However, in order to give the definition we first need to coinductively define
+ the well_formedness predicate, whose definition is more complex than the
+ previous one. *)
+coinductive well_formed': stream state → Prop ≝
+   wf_snil: ∀s. step s = None … → well_formed' (scons … s (snil …))
+ | wf_scons:
+    ∀hd1,hd2,tl.
+     step hd1 = Some … hd2 →
+      well_formed' (scons … hd2 tl) →
+       well_formed' (scons … hd1 (scons … hd2 tl)).
+
+(* Note: we could have equivalently defined well_formed' avoiding coinduction
+ by defining a recursive function to retrieve the n-th element of the stream,
+ if any. From now on we will stick to coinductive predicates only to show more
+ examples of usage of coinduction. In a formalization, however, it is always
+ better to explore several alternatives and see which ones work best for the
+ problem at hand. *)
+
+record trace': Type[0] ≝
+ { tr':> stream state
+ ; well_formed'': well_formed' tr'
+ }.
+
+(* The trace is diverging if every state is not final. Again, we show here
+ a coinductive definition. *)
+coinductive diverging': stream state → Prop ≝
+ mk_diverging': ∀hd,tl. diverging' tl → diverging' (scons … hd tl).
+
+(* The two coinductive definitions of diverging traces are equivalent.
+ To state the two results we first need a function to retrieve the head
+ from traces and diverging traces. *)
+definition head_of_streamseq: ∀A:Type[0]. streamseq A → A ≝
+ λA,s. match s with [ sscons hd _ ⇒ hd ].
+
+definition head_of_stream: ∀A:Type[0]. stream A → option A ≝
+ λA,s. match s with [ snil ⇒ None … | scons hd _ ⇒ Some … hd ].
+
+(* A streamseq can be extracted from a diverging stream using corecursion. *)
+let corec mk_diverging_trace_to_div_trace'
+ (s: stream state) : diverging' s → streamseq state ≝
+ match s return λs. diverging' s → streamseq state
+ with
+ [ snil ⇒ λabs: diverging' (snil …). ?
+ | scons hd tl ⇒ λH. sscons ? hd (mk_diverging_trace_to_div_trace' tl …) ].
+ [ cases (?:False) inversion abs #hd #tl #_ #abs' destruct
+ | inversion H #hd' #tl' #K #eq destruct @K ]
+qed.
+
+(* An expansion lemma will be needed soon. *)
+lemma mk_diverging_trace_to_div_trace_expansion:
+ ∀hd,tl,H. ∃K.
+  mk_diverging_trace_to_div_trace' (scons state hd tl) H =
+   sscons … hd (mk_diverging_trace_to_div_trace' tl K).
+ #hd #tl #H cases (eta_streamseq … (mk_diverging_trace_to_div_trace' ??)) /2/
+qed.
+
+(* CSC: BUG CHE APPARE NEL PROSSIMO LEMMA AL MOMENTO DELLA QED. IL DEMONE
+ SERVE PER DEBUGGARE *)
+axiom daemon: False.
+
+(* To complete the proof we need a final lemma: streamseqs extracted from
+ well formed diverging streams are well formed too. *) 
+let corec div_well_formed_co_mk_diverging_trace_to_div_trace (t : stream state) :
+ ∀H:diverging' t. div_well_formed_co (mk_diverging_trace_to_div_trace' t H) ≝
+ match t return λt. diverging' t → ?
+ with
+ [ snil ⇒ λabs. ?
+ | scons hd tl ⇒ λH. ? ].
+[ cases (?:False) inversion abs #hd #tl #_ #eq destruct
+| cases (mk_diverging_trace_to_div_trace_expansion … H) #H' #eq
+  lapply (sym_eq ??? … eq) #Req cases Req -Req -eq -H
+  cases tl in H';
+  [ #abs cases (?:False) inversion abs #hd #tl #_ #eq destruct
+  | -tl #hd2 #tl #H
+    cases (mk_diverging_trace_to_div_trace' … H) #H'
+    #eq lapply (sym_eq ??? … eq) #Req cases Req -Req
+    % [2: (*CSC: BIG BUG HERE*) cases daemon (* cases eq @div_well_formed_co_mk_diverging_trace_to_div_trace *)
+      | cases daemon ]]]
+qed.
+
+theorem diverging_trace_to_div_trace':
+ ∀t: trace'. diverging' t → ∃d: div_trace'.
+  head_of_stream … t = Some … (head_of_streamseq … d).
+ #t #H %
+ [ %{(mk_diverging_trace_to_div_trace' … H)}
+ | cases t in H; * normalize // #abs cases (?:False) inversion abs
+   [ #s #_ #eq destruct | #hd1 #hd2 #tl #_ #_ #eq destruct ]]
+ 
+  #n  lapply (well_formed t n)
+   lapply (H n) cases (tr t n) normalize [ * #abs cases (abs …) // ]
+   * #o #k #_ lapply (H (S n)) -H
+   cases (tr t (S n)) normalize
+   [ * #abs cases (abs …) // ] * #o' #k' #_ #EQ lapply (EQ … (refl …)) -EQ
+     normalize cases k normalize [ #abs destruct ] #hd #tl #EQ destruct -EQ
+     @step_next >e0 // ]
+ | lapply (H O) -H cases (tr t O) [ * #abs cases (abs …) // ] // ]
+qed.
+
+(* A stream can be extracted from a streamseq using corecursion. *)
+let corec stream_of_streamseq (A: Type[0]) (s: streamseq A) : stream A ≝
+ match s with [ sscons hd tl ⇒ scons … hd (stream_of_streamseq … tl) ].
+
+(* The proof that the resulting stream is diverging also need corecursion.*)
+let corec diverging_stream_of_streamseq (s: streamseq state) :
+ diverging' (stream_of_streamseq … s) ≝
+ match s return λs. diverging' (stream_of_streamseq … s)
+ with [ sscons hd tl ⇒ mk_diverging' … ].
+  mk_diverging' hd (stream_of_streamseq … tl) (diverging_stream_of_streamseq tl) ].
+ 
+
+theorem div_trace_to_diverging_trace':
+ ∀d: div_trace'. ∃t: trace'. diverging' t ∧
+  head_of_stream … t = Some … (head_of_streamseq … d).
+ #d %{(mk_trace' (stream_of_streamseq … d) …)}
+ [2: %
+   [  
+   [2: cases d * // ] #n % #abs destruct
+ | #n #s #EQ destruct lapply (div_well_formed d n) /2 by div_well_formed, next_step/ ]
+qed.
+
+
+(* ################## COME SPIEGARLO QUI? ####################### *)
+
+
+(*let corec stream_coind (A: Type[0]) (P: Prop) (H: P → Sum unit (A × P))
+ (p:P) : stream A ≝
+ match H p with
+ [ inl _ ⇒ snil A
+ | inr cpl ⇒ let 〈hd,p'〉 ≝ cpl in scons A hd (stream_coind A P H p') ]. *)
+
+(*lemma eta_streamseq:
+ ∀A:Type[0]. ∀s: streamseq A.
+  s = match s with [ sscons hd tl ⇒ sscons … hd tl ].
+ #A * //
+qed.
+
+lemma Rplus_streamseq_nf:
+ ∀xhd,xtl,yhd,ytl.
+  Rplus_streamseq (sscons … xhd xtl) (sscons … yhd ytl) =
+   sscons … (xhd + yhd) (Rplus_streamseq xtl ytl).
+ #xhd #xtl #yhd #ytl >(eta_streamseq Q (Rplus_streamseq …)) in ⊢ (??%?); //
+qed.*)
+
-- 
2.39.2